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A strong law for weighted sums of i.i.d. random variables. (English) Zbl 0833.60031
Let $$X$$, $$X_1, X_2, \dots, X_n, \dots$$ be i.i.d. random variables with $${\mathbf E}X = 0$$ and $$S_n = \sum^n_{i = 1} a_{in} X_n$$, where $$\{a_{in}\}_{i = 1(1)n}$$ is an array of constants. E.g., for the sequence $$\{S_n\}$$ of weighted sums a strong law is proved in the case $$\sup_{n} \root q \of {{1\over n} \sum^n_{i = 1} |a_{in}|^q} < \infty, q\in (1;\infty],$$ in the following manner: ${\mathbf E} |X|^p < \infty,\;p := {q\over q -1}, \text{ implies } {1\over n} S_n \to 0 \text{ a.s. (Theorem 1.1).}$ Also the case $$q = 1$$ is investigated. Extensions to more general normalizing sequences are given and necessary and sufficient conditions are discussed. Several well-known results are contained, e.g. in the case $$q = \infty$$, i.e. $$\text{sup} |a_{in}|< \infty$$ and $$p = 1$$, the result by B. D. Choi and S. H. Sung [Stochastic Anal. Appl. 5, 365-377 (1987; Zbl 0633.60049)].